Transmission and reception of channel state information

ABSTRACT

The present disclosure is related to reporting channel state information associated with a downlink channel in a wireless communication system.

TECHNICAL FIELD

The present disclosure relates to reporting channel state informationassociated with a downlink channel in a wireless communication system.

BACKGROUND ART

In order to overcome a frequency shortage problem, a next generationwireless communication system is being developed such that every basestation can use an entire available bandwidth. However, it may result ininter-cell interference since the same band width may be used at celledges. In order to overcome such problem, in the next generationwireless communication system, coordinated multi-pointtransmission/reception (CoMP) techniques may be considered.Particularly, a CoMP-joint processing (JP) technique among CoMPtechniques may enable a plurality of transmission points tosimultaneously transmit data to one user equipment. Accordingly, suchCoMP-JP technique may not cause interference and also provide atransmission point diversity effect, and therefore may improveperformance at cell edges.

In order to employ such technique (e.g., CoMP-JP), channel stateinformation may be required to be accurately reported in a downlinkchannel corresponding to a communication path from each transmissionpoint to user equipment. Furthermore, reporting of channel stateinformation may be required so as to not cause large feedback overhead.

DISCLOSURE OF INVENTION Technical Problem

An object of the present embodiment is to provide methods fortransmitting and receiving channel state information such that feedbackoverhead is reduced and also a reliability (or accuracy) of channelstate information is improved, and to provide user equipment and atransmission point therefor.

Technical Solution

In order to accomplish the above-described object, a method according toat least one embodiment may be provided for transmitting channel stateinformation in user equipment. The method may include extracting one ormore reference vectors having a high correlation with a channel stateinformation vector associated with a downlink; creating compressedvectors by compressing the reference vectors; and transmitting thecompressed vectors.

In accordance with another embodiment, user equipment may be provided.The user equipment may include a sparsity transform processor, acompression processor, and a transceiver. The sparsity transformprocessor may be configured to extract one or more reference vectorshaving a high correlation with a channel state information vectorassociated with a downlink. The compression processor may be configuredto create compressed vectors by compressing the reference vectors. Thetransceiver may be configured to transmit the compressed vectors.

In accordance with still another embodiment, a method may be providedfor receiving channel state information in a transmission point. Themethod may include receiving one or more compressed vectors from userequipment; creating decompressed vectors by decompressing the compressedvectors; extracting restoration vectors having a high correlation withthe decompressed vectors; and creating a channel information vector byadding the restoration vectors.

In accordance with still another embodiment, a transmission point may beprovided. The transmission point may include a transceiver, adecompression processor, and a channel state information extractionprocessor. The transceiver may be configured to receive one or morecompressed vectors from user equipment. The decompression processor maybe configured to create decompressed vectors by decompressing thecompressed vectors. The channel state information extraction processormay be configured to extract restoration vectors having a highcorrelation with the decompressed vectors, and to create a channelinformation vector by adding the restoration vectors.

Advantageous Effects

In case of reporting channel state information, the above-describedpresent embodiment may reduce feedback overhead and also improve areliability (or accuracy) of the channel state information.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a telecommunication system to which the presentembodiments may be applied.

FIG. 2 illustrates an exemplary system to which a CoMP-JP technique maybe applied.

FIG. 3 illustrates a structure of user equipment in accordance with atleast one embodiment.

FIG. 4 illustrates a method of transmitting channel state information(CSI) in the user equipment illustrated in FIG. 3.

FIG. 5 illustrates a structure of a transmission point in accordancewith at least one embodiment.

FIG. 6 illustrates a method of receiving channel state information (CSI)in the transmission point illustrated in FIG. 5.

MODE FOR CARRYING OUT THE INVENTION

Hereinafter, exemplary embodiments of the present invention will bedescribed with reference to the accompanying drawings. In the followingdescription, the same elements will be designated by the same referencenumerals although they are shown in different drawings. Furthermore, inthe following description of the present embodiment, a detaileddescription of known functions and configurations incorporated hereinwill be omitted when it may make the subject matter of the presentembodiment unclear.

FIG. 1 illustrates a telecommunication system to which the presentembodiments may be applied.

The telecommunication system may be widely used in order to provide avariety of communication services such as a voice service, a packet dataservice, and so forth.

Referring to FIG. 1, the telecommunication system may include userequipment (UE) 10 and transmission point 20. Herein, transmission point20 may perform uplink and downlink communications with user equipment10.

In the present specification, the term “user equipment (UE)” (e.g., userequipment 10) is used as a general concept that includes a user terminalin wireless communication. Accordingly, the user equipment (UE) shouldbe construed as a concept that includes a mobile station (MS), a userterminal (UT), a subscriber station (SS), and/or a wireless device in aglobal system for mobile communications (GSM), as well as user equipmentused in wideband code division multiple access (WCDMA), long termevolution (LTE), and/or high speed packet access (HSPA).

Generally, transmission point 20 or a cell may indicate a station thatcommunicates with user equipment 10. Such a transmission point may bereferred to as different terms, for example, a base station (BS), aNode-B, an evolved Node-B (eNB), a base transceiver system (BTS), anaccess point (AP), a relay node (RN), and the like.

In the present specification, transmission point 20 or the cell may beconstrued as an inclusive concept indicating a portion of an areacovered by a base station controller (BSC) in code division multipleaccess (CDMA), a Node-B in WCDMA, and the like. Furthermore,transmission point 20 or the cell may be construed as an inclusiveconcept indicating all types of devices capable of communicating withone user equipment. For example, transmission point 20 or the cell mayinclude a remote radio head (RRH) connected to a base station, a relaynode (RN), a sector of a macrocell, a site, a microcell (e.g., afemtocell, a picocell), or the like.

In the present specification, user equipment 10 and transmission point20 may be transmission/reception subjects, having an inclusive meaning,which are used to embody the technology and the technical conceptdisclosed herein, and may not be limited to a specific term or word.

FIG. 1 illustrates one user equipment (e.g., 10) and one transmissionpoint (e.g., 20). However, the present embodiment is not limitedthereto. More specifically, one transmission point (e.g., 20) maycommunicate with a plurality of user equipments (e.g., 10, and soforth). Alternatively, one user equipment (e.g., 10) may communicatewith a plurality of transmission points (e.g., 20, and so forth).

The communication system may use a variety of multiple access schemessuch as code division multiple access (CDMA), time division multipleaccess (TDMA), frequency division multiple access (FDMA), orthogonalfrequency division multiple access (OFDMA), OFDM-FDMA, OFDM-TDMA,OFDM-CDMA, and/or the like. Such multiple access schemes, however, arenot limited thereto. The present embodiment may be applied to such avariety of multiple access schemes.

In addition, in the case of an uplink transmission and a downlinktransmission, at least one of a time division duplex (TDD) scheme, afrequency division duplex (FDD) scheme, and a hybrid duplexing schememay be used. Herein, the TDD scheme may perform the uplink/downlinktransmissions using different times. The FDD scheme may perform theuplink/downlink transmissions using different frequencies. The hybridduplexing scheme may be a scheme which combines the FDD scheme and theTDD scheme. The present embodiment may be applied to the TDD scheme, theFDD scheme, and/or the hybrid duplexing scheme.

More specifically, the present embodiment may be applied in the field ofasynchronous wireless communications evolving to LTE and LTE-advanced(LTE-A) through GSM, WCDMA, and HSPA, and in the field of synchronouswireless communications evolving into CDMA, CDMA-2000, and UMB.Particularly, the present embodiment should not be construed as beinglimited to or restricted by a particular wireless communication field,and should be construed as including all technical fields to which thespirit of the present embodiment can be applied.

Referring to FIG. 1, user equipment 10 and transmission point 20 mayperform uplink and downlink communications.

Transmission point 20 may perform a downlink transmission to userequipment 10. Transmission point 20 may transmit a physical downlinkshared channel (PDSCH) which corresponds to a main physical channel usedfor unicast transmission. Furthermore, transmission point 20 maytransmit control channels such as a physical downlink control channel(PDCCH), a physical control format indicator channel (PCFICH), and/or aphysical HARQ indicator channel (PHICH). Herein, the PDCCH may be usedto transmit downlink control information, such as scheduling informationrequired for reception of PDSCH, and to transmit scheduling grantinformation for an uplink data channel (e.g., PUSCH) transmission. ThePCFICH may be used to transmit an indicator informing a division of aPDSCH region and a PDCCH region. The PHICH may be used for transmissionof ‘hybrid automatic repeat request (HARQ)’ acknowledgements in responseto an uplink transmission. Hereinafter, “transmit or receive a signalthrough a channel” may be referred to as the expression of “transmit orreceive a channel.”

Transmission point 20 may transmit reference signals on the downlink.Herein, the reference signals (i.e., downlink reference signals) mayinclude a cell-specific reference signal (CRS), a multicast/broadcastover single frequency network reference signal (MBSFN-RS), a UE-specificreference signal (may be referred to as a demodulation reference signal(DM-RS)), a positioning reference signal (PRS), and/or a channel stateinformation reference signal (CSI-RS).

User equipment 10 may perform an uplink transmission to transmissionpoint 20. User equipment 10 may transmit a physical uplink sharedchannel (PUSCH) which corresponds to a main physical channel used forunicast transmission. Furthermore, user equipment 10 may transmit aphysical uplink control channel (PUCCH) used for transmission of uplinkcontrol information (UCI) such as an HARQ acknowledgement, a channelstate report, a scheduling request, and so forth. Herein, the HARQacknowledgement may indicate whether a downlink transport block wassuccessfully received. The scheduling request may be a request for aresource allocation for an uplink data transmission.

User equipment 10 may transmit a demodulation reference signal (DRS) anda sounding reference signal (SRS) on an uplink.

In order to overcome a frequency shortage problem, a next generationwireless communication system is being developed such that everytransmission point can use an entire available bandwidth. However, itmay result in severe inter-cell interference since the same bandwidthmay be used at cell edges.

In order to overcome such problem, coordinated multi-pointtransmission/reception (CoMP) techniques may be considered. Such CoMPtechniques may be classified into a CoMP coordinatedscheduling/beamforming (CS/CB) technique and a CoMP-joint processing(JP) technique. According to the CoMP CS/CB technique, user equipmentmay communicate with one of a plurality of transmission points, and thetransmission points may be connected to each other to exchangeinformation on scheduling and beamforming.

FIG. 2 illustrates an exemplary system to which a CoMP-JP technique maybe applied. User equipment 10 (UE 1) at a cell edge may communicate witha plurality of transmission points (e.g., 20-1, 20-2). In other words,user equipment 10 may receive data from a plurality of transmissionpoints (e.g., 20-1, 20-2), and combine data signals received from theplurality of transmission points (e.g., 20-1, 20-2). Accordingly,performance at cell edges may be improved.

In order to perform a scheduling such that user equipment 10 can receivedata from a plurality of transmission points (e.g., 20-1, 20-2), statesof downlink channels from the plurality of transmission points (e.g.,20-1, 20-2) are required to be known. For this, schemes using at leastone of an implicit feedback, an explicit feedback, and an SRS may beapplied.

In a case of the implicit feedback, user equipment 10 may measure adownlink channel state based on a downlink reference signal(s) (e.g.,CRS, CSI-RS, etc.) transmitted from a transmission point(s) (e.g., 20-1and/or 20-2). Hereinafter, the terms “measure” and “measurement” areused as an inclusive concept including the terms “estimate” and“estimation.” User equipment 10 and the transmission point(s) (e.g.,20-1 and/or 20-2) may have the same predetermined codebook. Accordingly,user equipment 10 may obtain (or find) an index of the most similarvector to the measured channel state, from the codebook, and mayfeedback the obtained index to one or a plurality of transmission points(e.g., 20-1 and/or 20-2). Such implicit feedback scheme has an advantageof less feedback overhead. However, in a case of CoMP operations, aperformance improvement effect may not be large since only lessinformation is fed back.

In a case of the explicit feedback, user equipment 10 may measure adownlink channel state based on a downlink reference signal(s) (e.g.,CRS, CSI-RS, etc.) transmitted from a transmission point(s) (e.g., 20-1and/or 20-2). User equipment 10 may feedback information itself on themeasured channel state to the transmission point(s) (e.g., 20-1 and/or20-2). Such explicit feedback scheme has an advantage of enablingapplication of an improved transmission method by accurately measuringchannel states. However, such explicit feedback scheme may be lesseffective since feedback overhead associated with a channel matrix, achannel covariance matrix, eigenvectors, and eigenvalues is large.

The SRS may be transmitted on a symbol on an uplink from user equipment10 for a channel state measurement. The transmission point(s) (e.g.,20-1 and/or 20-2) may receive the SRS and measure an uplink channelstate using the received SRS. Furthermore, the transmission point(s)(e.g., 20-1 and/or 20-2) may measure a downlink channel state from theuplink channel state measured based on the SRS. Such channel statemeasurement using an SRS has an advantage of directly obtaining achannel state without an additional signal processing. However, in caseof FDD, a channel state measured on an uplink (i.e., an uplink channelstate) may be difficult to apply to scheduling for a downlink since thedownlink and uplink use different frequencies.

Hereinafter, the present embodiment may provide an explicit feedbackscheme enabling to reduce overhead.

FIG. 3 illustrates a structure of user equipment 300 in accordance withat least one embodiment. FIG. 4 illustrates a method of transmittingchannel state information (CSI) in user equipment 300 illustrated inFIG. 3.

Referring to FIG. 3, user equipment 300 may include transceiver 301,channel measurement processor 302, sparsity transform processor 303, andcompression processor 304.

Referring to FIG. 3 and FIG. 4, at step S401, transceiver 301 mayreceive a reference signal (e.g., CRS, CSI-RS, or the like) for channelmeasurement, from one or more transmission points. At step S402, channelmeasurement processor 302 may measure a downlink channel state based onthe reference signal which is received by transceiver 301. Furthermore,channel measurement processor 302 may create a channel vector includinginformation on the measured channel state.

At step S403, sparsity transform processor 303 may transform the channelvector created by channel measurement processor 302 into one or morereference vectors. In the case that a channel vector ({right arrow over(C)}) is a row vector having an l number of components, correlationbetween ‘each row of a sparse domain transformation matrix (F)’ and ‘thechannel vector ({right arrow over (C)})’ may be obtained by multiplyingthe sparse domain transformation matrix (F) having a size of l*l by thechannel vector ({right arrow over (C)}). Herein, ‘l’ may be a positiveinteger. In this case, one or more columns having a highest correlationmay be selected as a reference vector ({right arrow over (S)}) havingan/number of components.

Herein, the sparse domain transformation matrix (F) may be a squarematrix.

In at least one embodiment, the sparse domain transformation matrix (F)may be a unitary discrete Fourier transform (DFT) matrix having a sizeof l*l, as described in Formula 1 below.

$\begin{matrix}{{F = {\frac{1}{\sqrt{l}}\begin{bmatrix}1 & 1 & 1 & \; & 1 \\1 & \omega & \omega^{2} & \ldots & \omega^{({l - 1})} \\1 & \omega^{2} & \omega^{4} & \; & \omega^{2{({l - 1})}} \\\; & \vdots & \; & \ddots & \vdots \\1 & \omega^{l - 1} & \omega^{2{({l - 1})}} & \ldots & \omega^{{({l - 1})}^{2}}\end{bmatrix}}},{\omega = ^{{- j}\frac{2\; \pi}{l}}}} & \left\lbrack {{Formula}\mspace{14mu} 1} \right\rbrack\end{matrix}$

The unitary discrete Fourier transform (DFT) matrix of Formula 1 may bean example. Various types of unitary matrices may be used as the sparsedomain transformation matrix (F). If a unitary matrix is designed suchthat columns of the unitary matrix are orthogonal each other, channelinformation may be sustained even in a sparse domain, and therefore anaccuracy (or reliability) of feedback information may increase.

The number of selected reference vectors ({right arrow over (S)}) may bedetermined according to sparsity levels which are determined betweenuser equipment and a transmission point. For example, in the case thatthe sparsity level is “1”, in a sparse domain transformation matrix (F),one column vector having the highest correlation with a channel vector({right arrow over (C)}) may be selected as a reference vector ({rightarrow over (S)}).

In the case that the sparsity level is “2” or higher, in a sparse domaintransformation matrix (F), two or more column vectors having relativelyhigher correlations with a channel vector ({right arrow over (C)}) maybe selected as reference vectors ({right arrow over (S)}). Furthermore,a weight (i.e., a weight value) for each reference vector ({right arrowover (S)}) may be obtained. In this case, weights of reference vectorsmay be obtained such that a weight for each reference vector ({rightarrow over (S)}) is in proportion to a correlation with each referencevector ({right arrow over (S)}) and a summation of weights is “1”.Alternatively, a weight for each reference vector ({right arrow over(S)}) may be obtained by applying two or more selected reference vectors({right arrow over (S)}) to a least square method (LSM).

In some embodiments, in the case that the sparsity level is “2” orhigher, one column vector having the highest correlation with a channelvector ({right arrow over (C)}) may be determined in a sparse domaintransformation matrix (F), and a weight for the one column vector may beobtained using a least square method (LSM). After subtracting a productof the column vector and its weight from the channel vector ({rightarrow over (C)}), one column vector having the highest correlation and acorresponding weight may be obtained from the result vector(s) again.Such operations may be repetitively performed until column vectors andweights as many as the number corresponding to the sparsity level areobtained.

In other embodiments, in the case that the sparsity level is “2” orhigher, all column vectors in a sparse domain transformation matrix (F)may be determined, and weights for the all column vectors may beobtained using a least square method (LSM). Accordingly, column vectorsand weights as many as the number corresponding to the sparsity levelmay be obtained.

The weights (i.e., weight values) may be managed as separate data.Furthermore, a reference vector ({right arrow over (S)}) may be updatedas a result value created by multiplying the reference vector ({rightarrow over (S)}) by a corresponding weight.

A sparse domain transformation matrix (F) may be a predetermined matrix.Alternatively, a sparse domain transformation matrix (F) may bedetermined by a transmission point. In this case, the determined sparsedomain transformation matrix (F) may be transferred to user equipmentfrom the transmission point (i) through a higher-layer signal (e.g.,radio resource control (RRC) signal) or (ii) through a downlink controlchannel (e.g., PDCCH). Alternatively, a sparse domain transformationmatrix (F) may be determined by user equipment. In this case, thedetermined sparse domain transformation matrix (F) may be transferred toa transmission point from the user equipment through an uplink controlchannel (e.g., PUCCH).

A sparsity level may be a predetermined value. Alternatively, a sparsesparsity level may be determined by a transmission point. In this case,the determined sparsity level may be transferred to user equipment fromthe transmission point (i) through a higher-layer signal (e.g., RRCsignal) or (ii) through a downlink control channel (e.g., PDCCH).Alternatively, a sparsity level may be determined by user equipment. Inthis case, the determined sparsity level may be transferred to atransmission point from the user equipment through an uplink controlchannel (e.g., PUCCH).

At step S404, compression processor 304 may create (or extract) one ormore compressed vectors ({right arrow over (v)}) by compressing the oneor more reference vectors ({right arrow over (S)}) obtained by sparsitytransform processor 303.

An optimization matrix (O) may be prepared for a compression of thereference vectors ({right arrow over (S)}). The optimization matrix (O)may be a matrix which has a size of 2^(p)*l (where l≦2^(p)(2^(p)+1) andp is a prime number) and whose rows are orthogonal to each other. Forexample, in the case that the prime number ‘p’ is 2 (i.e., p=2), theoptimization matrix (O) may have a size of 4*l (where l≦20). In the casethat the prime number ‘p’ is 3 (i.e., p=3), the optimization matrix (O)may have a size of 8*l (where l≦72). Herein, the optimization matrix (O)may be formed by a product (i.e., O=CF) of a compression matrix (C) andsparse domain transformation matrix (F). In at least one embodiment, anoptimization matrix (O) may be formed using a Hadamard matrix. In otherembodiments, an optimization matrix (O) may be formed using aGrassmannian manifold design scheme. Herein, a matrix according to theGrassmannian manifold design scheme may have an optimal characteristicsince correlations between all columns are the same. However, in case ofthe Grassmannian manifold design scheme, only a specific size of matrixmay be formed, and each column is required to be heuristicallydetermined.

Accordingly, a compression matrix (C) may be determined by Formula 2below.

C=OF ^(H)  [Formula 2]

Herein, ‘H’ represents a Hermitian operator. The above-describedoptimization matrix (O) may be determined such that cross correlationsbetween all columns of the compression matrix (C) are maximized and acompression rate is relatively maximal.

In Formula 2, the compression matrix (C) may have a size of 2^(p)*l.

An optimization matrix (O) and/or a compression matrix (C) may bepredetermined values. Alternatively, an optimization matrix (O) and/or acompression matrix (C) may be determined by a transmission point. Inthis case, the determined optimization matrix (O)/compression matrix (C)or indicators therefor may be transferred to user equipment from thetransmission point (i) through a higher-layer signal (e.g., RRC signal)or (ii) through a downlink control channel (e.g., PDCCH).

As shown in the following Formula 3, a compressed vector ({right arrowover (v)}) may be calculated by multiplying a compression matrix (C) byone or more reference vectors (S).

{right arrow over (v)}=C·{right arrow over (S)}  [Formula 3]

The compressed vector {right arrow over (v)}calculated by theabove-described Formula 3 may be a row vector formed by a 2^(p) numberof components. Compressed vectors {right arrow over (v)} as many as thenumber corresponding to a sparsity level may be calculated by Formula 3.

In addition, at step S405, transceiver 301 may transmit one or morecompressed vectors {right arrow over (v)}created in compressionprocessor 304, to a transmission point.

Example 1-1

Description will be given under the assumption that a channel vector({right arrow over (C)}) to be transmitted by user equipment 300 is amatrix having 20 components, as follows. Herein, the channel vector({right arrow over (C)}) may include channel information.

{right arrow over (C)}=[1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1−1]^(T)/√{square root over (20)}

A unitary DFT matrix having a size of 20*20 as shown in Formula 4 belowmay be used as a sparse domain transformation matrix (F).

$\begin{matrix}{{F = {\frac{1}{\sqrt{20}}\begin{bmatrix}1 & 1 & 1 & \; & 1 \\1 & \omega & \omega^{2} & \ldots & \omega^{19} \\1 & \omega^{2} & \omega^{4} & \; & \omega^{38} \\\; & \vdots & \; & \ddots & \vdots \\1 & \omega^{19} & \omega^{38} & \ldots & \omega^{361}\end{bmatrix}}},{\omega = ^{{- j}\frac{2\; \pi}{20}}}} & \left\lbrack {{Formula}\mspace{14mu} 4} \right\rbrack\end{matrix}$

In the case that a sparsity level is determined as “1”, among columns ofthe above-described sparse domain transformation matrix (F), a columnhaving the highest correlation with the channel vector ({right arrowover (C)}) may be found, and be determined as a reference vector ({rightarrow over (S)}). In the above-described example, a result vectoraccording to a product of the sparse domain transformation matrix (F)and the channel vector ({right arrow over (C)}) may be as below.

F{right arrow over (C)}=[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0]^(T)

From the result vector, it may be known that the 11^(th) column vectorof the sparse domain transformation matrix (F) has the highestcorrelation with the channel vector ({right arrow over (C)}).Accordingly, a reference vector ({right arrow over (S)}) determined asthe 11^(th) column vector of the sparse domain transformation matrix (F)may be as below.

{right arrow over (S)}=[1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1−1]^(T)/√{square root over (20)}

In the case that the prime number ‘p’ is 2 (i.e., p=2), an optimizationmatrix (O) may be a matrix having a size of 4*20 as shown in Formula 5below.

$\begin{matrix}{O = {{CF} = \left\lbrack {I_{4}{{\frac{1}{2}D_{1}H_{4}}{{\frac{1}{2}D_{2}H_{4}}{{\frac{1}{2}D_{3}H_{4}}{\frac{1}{2}D_{4}H_{4}}}}}} \right\rbrack}} & \left\lbrack {{Formula}\mspace{14mu} 5} \right\rbrack\end{matrix}$

In Formula 5, I₄ represents a unit matrix having a size of 4*4. H₄represents a 4*4 Hadamard matrix (i.e., Hadamard matrix having a size of4*4) as below.

$H_{4} = \begin{bmatrix}1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1\end{bmatrix}$

A Hadamard matrix may be a square matrix including values of “1” or“−1”. Furthermore, the Hadamard matrix may have a characteristic thateach column of the Hadamard matrix is orthogonal to each other.

Meanwhile, in the case that the prime number ‘p’ is 3 (i.e., p=3), an8*8 Hadamard matrix (H₈) may be used for configuration of anoptimization matrix (O). H₈ may be as follows.

$H_{8} = \begin{bmatrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\1 & {- 1} & 1 & {- 1} & 1 & {- 1} & 1 & {- 1} \\1 & 1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} \\1 & {- 1} & {- 1} & 1 & 1 & {- 1} & {- 1} & 1 \\1 & 1 & 1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} \\1 & {- 1} & 1 & {- 1} & {- 1} & 1 & {- 1} & 1 \\1 & 1 & {- 1} & {- 1} & {- 1} & {- 1} & 1 & 1 \\1 & {- 1} & {- 1} & 1 & {- 1} & 1 & 1 & {- 1}\end{bmatrix}$

D_(k), (k=1, 2, 3, or 4) may be a value indicating an optimal phaserotation parameter of a Hadamard matrix, and be determined such thatcross correlations between all columns of a compression matrix (C) aremaximized and a compression rate is relatively maximal. In case ofExample 1, D_(k) may be matrices shown in Formula 6 below.

$\begin{matrix}{{D_{1} = {{diag}\; \left( \begin{bmatrix}^{j\frac{\pi}{4}} & ^{j\frac{\pi}{4}} & ^{j\frac{\pi}{4}} & ^{j\frac{\pi}{4}}\end{bmatrix} \right)}},{D_{2} = {{diag}\; \left( \begin{bmatrix}^{j\frac{\pi}{4}} & ^{j\frac{3\pi}{4}} & ^{j\frac{3\pi}{4}} & ^{j\frac{5\pi}{4}}\end{bmatrix} \right)}},{D_{3} = {{diag}\; \left( \begin{bmatrix}^{j\frac{\pi}{4}} & ^{{j3}\frac{\pi}{4}} & ^{j\frac{\pi}{4}} & ^{j\frac{7\pi}{4}}\end{bmatrix} \right)}},{D_{4} = {{diag}\; \left( \begin{bmatrix}^{j\frac{\pi}{4}} & ^{j\frac{\pi}{4}} & ^{j\frac{3\pi}{4}} & ^{j\frac{7\pi}{4}}\end{bmatrix} \right)}}} & \left\lbrack {{Formula}\mspace{14mu} 6} \right\rbrack\end{matrix}$

A compression matrix (C) may be a 4*20 matrix (i.e., a matrix having asize of 4*20) determined by Formula 2 (C=OF^(H)). In this case, acompressed vector ({right arrow over (v)}) created by a compressionoperation using the compression matrix (C) may be as below.

{right arrow over (v)}=C·{right arrow over (S)}=0.3536×[1+j−1+j−1−j−1+j]^(T)

According to the above-described scheme, the channel vector {right arrowover (C)}having 20 components may be compressed to the compressed vector({right arrow over (v)}) having four components. The compressed vector{right arrow over (v)}having four components may be transmitted fromuser equipment to a transmission point. Accordingly, feedback overheadmay be reduced.

Example 1-2

Description will be given under the assumption that (i) a channel vector{right arrow over (C)} to be transmitted by user equipment 300 is amatrix having 20 components, and (ii) the unitary DFT matrix defined byFormula 4 is used as a sparse domain transformation matrix (F). Herein,the channel vector {right arrow over (C)} may include channelinformation. In this case, it may be assumed that a result vectoraccording to a product of the sparse domain transformation matrix (F)and the channel vector {right arrow over (C)} is as below.

F{right arrow over (C)}=[0.8 0 0 0 0 0 0 0 0 0 0.2 0 0 0 0 0 0 0 00]^(T)

In the case that the sparsity level is determined as “2”, two columnvectors (e.g., the 1^(st) column vector and the 11^(th) column vector)having higher correlations in the sparse domain transformation matrix(F) may be selected as two reference vectors (e.g., {right arrow over(s₁)} and {right arrow over (s₂)}). Weights for the reference vectors(e.g., {right arrow over (s₁)} and {right arrow over (s₂)}) may bedetermined as “0.8” and “0.2”, respectively. Thereafter, each compressedvector (e.g., {right arrow over (v₁)} or {right arrow over (v₂)}) may becalculated by compressing each reference vector (e.g., {right arrow over(s₁)} or {right arrow over (s₂)}) by a compression matrix (C). Herein,each compressed vector (e.g., {right arrow over (s₁)} and {right arrowover (s₂)}) may have four components. In addition, the compressedvectors (e.g., {right arrow over (v₁)} or {right arrow over (v₂)}) eachof which has four components, and the corresponding weights (e.g., 0.8and 0.2) may be transmitted from user equipment to a transmission point.

Alternatively, the 1^(st) column vector and the 11^(th) column vector ofthe sparse domain transformation matrix (F) may be multiplied by theweights 0.8 and 0.2, respectively. In this case, the result vectors maybe determined as reference vectors (e.g., {right arrow over (s₁)} or{right arrow over (s₂)}). Thereafter, each compressed vector ({rightarrow over (v₁)} or {right arrow over (v₂)}) having four components maybe calculated by compressing each reference vector (e.g., {right arrowover (s₁)} or {right arrow over (s₂)}) by a compression matrix (C). Inaddition, the compressed vectors (e.g., {right arrow over (v₁)} or{right arrow over (v₂)}) each of which has four components may betransmitted from user equipment to a transmission point.

Example 2-1

In the case that channel information to be transmitted by user equipment300 includes an l number of components, a channel vector ({right arrowover (C)}) may be configured with (i) an l number of channel informationcomponents and (ii) a [2^(p)(2^(p)+1)−l] number of ‘0’ components.Herein, l may be less than 2^(p)(2^(p)+1). For example, in the case thatthe channel information includes 16 components (l=16), it may be assumedthat the prime number ‘p’ is 2 (i.e., p=2) and a channel vector ({rightarrow over (C)}) is the following matrix having 20 components.

{right arrow over (C)}=[1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 0 0 00]^(T)/√{square root over (20)}

In this case, a sparse domain transformation matrix (F) having a size of20*20 may be a matrix defined by Formula 4 of Example 1, and anoptimization matrix (O) having a size of 4*20 may be a matrix defined byFormula 5 of Example 1. One or more reference vectors ({right arrow over(S)}) having a high correlation with the channel vector {right arrowover (C)} may be extracted from the sparse domain transformation matrix(F) defined by Formula 4. Herein, each reference vector ({right arrowover (S)}) may have 20 components. Thereafter, each reference vector({right arrow over (S)}) having 20 components may compressed to acompressed vector ({right arrow over (v)}) having four components, byperforming a compression operation using a compression matrix (e.g.,C=OF^(H)). Herein, the compression matrix (e.g., C=OF^(H)) may beobtained from the sparse domain transformation matrix (F) defined byFormula 4 and the optimization matrix (O) defined by Formula 5.

Example 2-2

In the case that channel information to be transmitted by user equipment300 includes an l number of components, a channel vector ({right arrowover (C)}) may be configured with only an l number of channelinformation components. Herein, l may be less than 2^(p)(2^(p)+1). Forexample, in the case that the channel information includes 16components, it may be assumed that the prime number ‘p’ is 2 (i.e., p=2)and a channel vector ({right arrow over (C)}) is the following matrixhaving 16 (<20) components.

{right arrow over (C)}=[1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1−1]^(T)/√{square root over (16)}

In this case, the following DFT matrix may be used as a sparse domaintransformation matrix (F) having a size of 16*16.

${F = {\frac{1}{\sqrt{16}}\begin{bmatrix}1 & 1 & 1 & \; & 1 \\1 & \omega & \omega^{2} & \ldots & \omega^{15} \\1 & \omega^{2} & \omega^{4} & \; & \omega^{30} \\\; & \vdots & \; & \ddots & \vdots \\1 & \omega^{15} & \omega^{30} & \ldots & \omega^{225}\end{bmatrix}}},{\omega = ^{{- j}\frac{2\; \pi}{16}}}$

One or more reference vectors ({right arrow over (S)}) having arelatively high correlation with the channel vector ({right arrow over(C)}) may be extracted from the sparse domain transformation matrix (F).Herein, each reference vector ({right arrow over (S)}) may have 16components.

A matrix having a size of 4*16 may be selected as an optimization matrix(O). The 4*16 optimization matrix (O) may be formed by selectingarbitrary 16 columns in a 4*20 matrix (i.e., a matrix having a size of4*20) defined by Formula 5.

For example, an optimization matrix (O) having a size of 4*16 may beformed by excluding a unit matrix (I₄) from the matrix defined byFormula 5. In other words, the optimization matrix (O) having a size of4*16 may be

$O = {{CF} = {\left\lbrack {{\frac{1}{2}D_{1}H_{4}}{{\frac{1}{2}D_{2}H_{4}}{{\frac{1}{2}D_{3}H_{4}}{\frac{1}{2}D_{4}H_{4}}}}} \right\rbrack.}}$

Alternatively, an optimization matrix (O) having a size of 4*16 may beformed by excluding one Hadamard matrix from the matrix defined byFormula 5. In this case, for example, the optimization matrix (O) havinga size of 4*16 may be

$O = {{CF} = {\left\lbrack {I_{4}{{\frac{1}{2}D_{1}H_{4}}{{\frac{1}{2}D_{2}H_{4}}{\frac{1}{2}D_{3}H_{4}}}}} \right\rbrack.}}$

Meanwhile, an compression matrix (e.g., C=OF^(H)) having a size of 4*16may be obtained using the sparse domain transformation matrix (F) havinga size of 16*16 and the optimization matrix (O) having a size of 4*16.Furthermore, a compressed vector ({right arrow over (v)}) having fourcomponents may be obtained from the compression matrix (C) having a sizeof 4*16 and the reference vector ({right arrow over (S)}) having 16components.

FIG. 5 illustrates a structure of a transmission point (e.g.,transmission point 500) in accordance with at least one embodiment. FIG.6 illustrates a method of receiving channel state information (CSI) inthe transmission point (e.g., transmission point 500) illustrated inFIG. 5.

Referring to FIG. 5, transmission point 500 may include transceiver 501,decompression processor 502, and channel state information (CSI)extraction processor 503.

At step S601, transceiver 501 may receive one or more compressed vectors({right arrow over (v′)}) from user equipment. The received compressedvectors ({right arrow over (v′)}) may be ‘transmitted compressedvectors’ ({right arrow over (v′)}) (i.e., compressed vectors ({rightarrow over (v′)}) transmitted by the user equipment) to whichnoise/interference effects of channels are added. The number ofcompressed vectors ({right arrow over (v′)}) may be determined accordingto a sparsity level. Each compressed vectors ({right arrow over (v′)})may have a 2^(p) number of components. Herein, ‘p’ may be a primenumber. In the case that the sparsity level is 2 or higher, and weightsand compressed vectors are separately transmitted from user equipment,transceiver 501 may further receive weight information.

At step S602, decompression processor 502 may decompress each compressedvector ({right arrow over (v′)}). More specifically, decompressionprocessor 502 may create one or more decompressed vectors ({right arrowover (d)}) by multiplying a Hermitian matrix (O^(H)) of a referencematrix (O) used in user equipment by compressed vector(s) ({right arrowover (v′)}), as shown in Formula 7 below. Herein, the decompressedvector ({right arrow over (d)}) may have an/number of components.

{right arrow over (d)}=O ^(H) ·{right arrow over (v′)}  [Formula 7]

Channel state information (CSI) extraction processor 503 may obtain acorrelation between each column of a sparse domain transformation matrix(F) and the decompressed vector(s) ({right arrow over (d)}), bymultiplying the sparse domain transformation matrix (F) used in userequipment by the decompressed vector(s) ({right arrow over (d)}). Inthis case, one column having the highest correlation may be selected asa restoration vector ({right arrow over (r)}) having an l number ofcomponents. In the same manner, a restoration vector ({right arrow over(r)}) may be determined from each of one or more decompressed vectors({right arrow over (d)}). Each restoration vector ({right arrow over(r)}) may have an l number of components.

At step S603, a channel information vector ({right arrow over (CI)})including channel state information (CSI) may be obtained based on oneor more restoration vectors ({right arrow over (r)}). In the case that asparsity level is ‘1’, the channel information vector ({right arrow over(CI)}) may be the same as the restoration vector ({right arrow over(r)}). In the case that a sparsity level is 2 or higher, and compressedvectors ({right arrow over (v′)}) are created considering weights inuser equipment, the channel information vector ({right arrow over (CI)})may be sum of the restoration vectors ({right arrow over (r)}). In thecase that (i) a sparsity level is 2 or higher, (ii) compressed vectors({right arrow over (v′)}) are created without considering weights inuser equipment, and (iii) weights are separately transmitted,restoration vectors ({right arrow over (r)}) may be multiplied byweights. In this case, the channel information vector ({right arrow over(CI)}) may be sum of the multiplication results (i.e., sum of productsof restoration vectors ({right arrow over (r)}) and weights).

In the above-described method, correlation between each column vector ofa reference matrix (O) may be lowest. Accordingly, a probability ofextracting a column vector of a sparse domain transformation matrix (F)corresponding to a channel vector transformed into a sparse domain maybe maximal.

Meanwhile, steps S602 and S603 described above may be performed as asingle step. A vector having an l number of components may be obtainedby multiplying a Hermitian matrix (C^(H)) of a compression matrix(C=OF^(H)) used in user equipment by a compressed vector ({right arrowover (v′)}). A correlation between each column of a Hermitian matrix(C^(H)) of a compression matrix (C=OF^(H)) and a compressed vector({right arrow over (v′)}) may be obtained from the obtained vectorhaving an l number of components. In this case, a column having thehighest correlation may be selected, and a column of a sparse domaintransformation matrix (F) corresponding to the selected column (i.e.,the column having the highest correlation) may be selected as arestoration vector ({right arrow over (r)}). For example, if acorrelation between the 1^(st) column of a Hermitian matrix (C^(H)) of acompression matrix (C) and a compressed vector ({right arrow over (v′)})is highest, the 1^(st) column of a sparse domain transformation matrix(F) may be selected as a restoration vector ({right arrow over (r)}).

In the above-described manner, in case of one or more compressed vectors({right arrow over (v′)}), a restoration vector ({right arrow over (r)})may be determined for each compressed vector ({right arrow over (v′)}).At step S603, a channel information vector ({right arrow over (CI)}) maybe obtained from one or more restoration vectors ({right arrow over(r)}). More specifically, in the case that a sparsity level is ‘1’ and,therefore, there is a single compressed vector ({right arrow over(v′)}), the channel information vector ({right arrow over (CI)}) may bedetermined as one restoration vector ({right arrow over (r)}). In thecase that a sparsity level is higher than ‘1’ and therefore there are aplurality of compressed vectors ({right arrow over (v′)}), the channelinformation vector ({right arrow over (CI)}) may be determined as a sumof restoration vectors ({right arrow over (r)}). Information included inthe determined channel information vector ({right arrow over (CI)}) maybe used when a transmission point (e.g., transmission point 500)performs a downlink scheduling.

For example, in the case that it is assumed that noise and interferenceeffects of channels is negligible, a compressed vector ({right arrowover (v′)}) received by a transmission point may be the same as acompressed vector ({right arrow over (v)}) transmitted by userequipment. If Formula 2 and Formula 3 are applied to Formula 7, thefollowing Formula 8 may be induced.

{right arrow over (d)}=O ^(H) ·{right arrow over (v′)}=O ^(H) OF ^(H){right arrow over (S)}=F ^(H) {right arrow over (S)}  [Formula 8]

Accordingly, in this case, a restoration vector ({right arrow over (r)})may be the same as a reference vector ({right arrow over (S)}) of userequipment. Herein, the restoration vector ({right arrow over (r)}) maybe a column vector having the highest correlation with a decompressedvector ({right arrow over (d)}), among column vectors of a sparse domaintransformation matrix (F). In such manners, the reference vector ({rightarrow over (S)}) in the user equipment may be restored in a transmissionpoint.

For example, in the case that a sparsity level is ‘1’, the prime number‘p’ is 2 (i.e., p=2), and a channel vector ({right arrow over (C)}) isconfigured with 20 components, it may be assumed that a compressedvector ({right arrow over (v′)}) received by a transmission point (e.g.,transmission point 500) is 0.3536[1+j−1+j−1−j−1+j]^(T).

In the case that a reference matrix (O) used in user equipment is amatrix described in Formula 5, a decompressed vector ({right arrow over(d)}) according to Formula 7 may be calculated as [1 −1 1 −1 1 −1 1 −1 1−1 1 −1 1 −1 1 −1 1 −1 1 −1]^(T)/√{square root over (20)}.

In the case that a sparse domain transformation matrix (F) used userequipment is a matrix described in Formula 4, a restoration vector({right arrow over (r)}) may be [1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −1 1 −11 −1 1 −1]^(T)/√{square root over (20)}. Herein, the restoration vector({right arrow over (r)}) may be a column vector having the highestcorrelation with the above-described decompressed vector ({right arrowover (d)}), among column vectors of the sparse domain transformationmatrix (F).

A channel information vector ({right arrow over (CI)}) may be determinedas [1 −1 1 −1 1 −1 1 −1 1 −1 1−1 1−1 1−1 1−1]^(T)/√{square root over(20)} since a sparsity level is ‘1’. A transmission point may extractdownlink channel state information from the determined channelinformation vector ({right arrow over (CI)}). A downlink scheduling maybe performed based on the extracted downlink channel state information.

Methods of transmitting and/or receiving a channel state according tothe present embodiment described above may be embodied in the form ofprograms, and be recorded in a computer-readable recording medium.

A program recorded in a recording medium for implementation of a channelstate transmission method according to at least one present embodimentmay execute a function of transforming sparsity of a channel stateinformation vector, a function of compressing the sparsity-transformedvector, and so forth. A program recorded in a recording medium forimplementation of a channel state reception method according to at leastone present embodiment may execute a function of decompressing areceived compressed vector, a function of extracting a vector having thehighest correlation with the decompressed vector, and so forth.

The above-described program may include program codes coded usingcomputer languages such that a computer can read a program recorded in arecording medium and execute functions embodied by the program. Herein,the computer languages may be read through device interfaces by aprocessor (e.g., CPU) of the computer, and include C, C++, JAVA, amachine language, and/or the like.

Such codes may include function codes associated with functions definingthe above-described capabilities (or functions). Furthermore, such codesmay include execution-related control codes which are required toexecute the above-described capabilities (or functions) according to apredetermined procedure by a computer processor.

Further, such codes may further include codes associated with a memoryreference. Herein, the codes associated with the memory reference mayindicate a memory position (i.e., a position in an internal or externalmemory of a computer, for example, address) at which additionalinformation or media required to execute the above-describedcapabilities (or functions) by a computer processor can be referred.

Furthermore, in the case that a processor of a computer is required tocommunicate with other remote computers or servers in order to executethe above-described capabilities (or functions), such codes may furtherinclude codes associated with communications. Herein, the codesassociated with communications may indicate a communication scheme(i.e., how to communicate with other remote computers or servers) and/orcommunication objects (e.g., information or media to betransmitted/received), when the processor of the computer communicateswith the other remote computers or servers using a correspondingcommunication module.

Examples of the computer-readable recording medium to record theabove-described program may include a read-only memory (ROM), arandom-access memory (RAM), CD-ROMs, magnetic tapes, floppy disks,optical data storage devices, and the like. Furthermore, thecomputer-readable recording medium may include a medium embodied in theform of carrier waves (such as data transmission through the Internet).

The computer-readable recording medium may also be distributed overnetwork coupled computer systems such that computer-readable codes arestored and executed in a distributed manner.

Also, functional programs, codes, and code segments for accomplishingthe present invention may be easily construed or modified by programmersskilled in the art to which the present invention pertains, in view ofconsidering a system environment of a computer which can read arecording medium and execute programs.

As described above, since the technical idea of the present invention isdescribed by exemplary embodiments, various forms of substitutions,modifications and alterations may be made by those skilled in the artfrom the above description without departing from essential features ofthe present invention. Therefore, the embodiments disclosed in thepresent invention are intended to illustrate the technical idea of thepresent invention, and the scope of the present invention is not limitedby the embodiment. The scope of the present invention shall be construedon the basis of the accompanying claims in such a manner that all of thetechnical ideas included within the scope equivalent to the claimsbelong to the present invention.

CROSS-REFERENCE TO RELATED APPLICATION

The present application claims priority under 35 U.S.C. §119(a) toKorean Patent Application No. 10-2011-0071096 (filed on Jul. 18, 2011),which is hereby incorporated by reference in its entirety. In addition,the present application claims priority in countries, other than U.S.,with the same reason based on the Korean Patent Applications, which arehereby incorporated by reference in their entirety.

1. A method of transmitting channel state information in user equipmentincluding a transceiver, the method comprising: extracting one or morereference vectors having a high correlation with a channel stateinformation vector associated with a downlink; creating one or morecompressed vectors by compressing the one or more reference vectors; andtransmitting, by the transceiver, the one or more compressed vectors. 2.The method of claim 1, wherein: the correlation is calculated bymultiplying a square matrix by the channel state information vector,where columns of the square matrix are orthogonal to each other; and thereference vectors are column vectors having a high correlation in thesquare matrix.
 3. The method of claim 2, wherein: the channel stateinformation vector includes an l number of components indicating thechannel state information, where l is a positive integer; and the squarematrix is a matrix having a size of l*l.
 4. The method of claim 2,wherein: the channel state information vector includes (i) an l numberof components indicating the channel state information, where l is apositive integer, and (ii) a 2^(p)(2^(p)+1)−l number of components notindicating the channel state information, where p is a prime number; andthe square matrix is a matrix having a size of2^(p)(2^(p)+1)*2^(p)(2^(p)+1).
 5. The method of claim 1, wherein: thecompressed vectors are calculated by multiplying a compression matrix bythe reference vectors; and the compression matrix is a matrix having asize of 2^(p)*l, where l is a positive integer satisfyingl≦2^(p)(2^(p)+1) and p is a prime number, in a case that the referencevectors have an l number of components.
 6. User equipment comprising: asparsity transform processor configured to extract one or more referencevectors having a high correlation with a channel state informationvector associated with a downlink; a compression processor configured tocreate one or more compressed vectors by compressing the one or morereference vectors; and a transceiver configured to transmit the one ormore compressed vectors.
 7. The user equipment of claim 6, wherein: thecorrelation is calculated by multiplying a square matrix by the channelstate information vector, where columns of the square matrix areorthogonal to each other; and the reference vectors are column vectorshaving a high correlation in the square matrix.
 8. The user equipment ofclaim 7, wherein: the channel state information vector includes an lnumber of components indicating the channel state information, where lis a positive integer; and the square matrix is a matrix having a sizeof l*l.
 9. The user equipment of claim 7, wherein: the channel stateinformation vector includes (i) an l number of components indicating thechannel state information, where l is a positive integer, and (ii) a2^(p)(2^(p)+1)−l number of components not indicating the channel stateinformation, where p is a prime number; and the square matrix is amatrix having a size of 2^(p)(2^(p)+1)*2^(p) (2^(p)+1).
 10. The userequipment of claim 6, wherein: the compressed vectors are calculated bymultiplying a compression matrix by the reference vectors; and thecompression matrix is a matrix having a size of 2^(p)*l, where l is apositive integer satisfying l≦2^(p)(2^(p)+1) and p is a prime number, ina case that the reference vectors have an l number of components.
 11. Amethod of receiving channel state information in a transmission pointincluding a transceiver, the method comprising: receiving, by thetransceiver, one or more compressed vectors from user equipment;creating one or more decompressed vectors by decompressing the one ormore compressed vectors; extracting one or more restoration vectorshaving a high correlation with the one or more decompressed vectors; andcreating a channel information vector by adding the one or morerestoration vectors.
 12. The method of claim 11, wherein: thedecompressed vectors are calculated by multiplying a Hermitian matrix ofa reference matrix by the compressed vectors, and the reference matrixhas a low cross correlation between column vectors.
 13. The method ofclaim 12, wherein: the compressed vectors have a 2^(p) number ofcomponents, where p is a prime number; and the Hermitian matrix of thereference matrix is a matrix having a size of l*2^(p), where‘l’represents the number of the channel state information, and is apositive integer satisfying l≦2^(p)(2^(p)+1).
 14. The method of claim11, wherein: the correlation is calculated by multiplying a squarematrix by the decompressed vectors, where columns of the square matrixare orthogonal to each other; and the restoration vectors are columnvectors having a highest correlation in the square matrix. 15-18.(canceled)